How far back in history do you want to go? Plenty of battles within Christianity in British history. E.g. King Henry VIII didn't accept the authority of the pope and led the Church of England away from the Roman Catholic Church. https://en.wikipedia.org/wiki/Church_of_England#Separation_f...
A few generations later, things got pretty dicey for the remaining Catholics:
England’s Elizabethan Catholics were public
enemy number one. Their Masses were banned
and their priests were executed.
To the best of my knowledge, that didn't happen. The British Empire in its quests wasn't concerned about anti-Catholicism and disdain for the authority of the Pope, it was primarily driven by political/financial gains. On the other hand, for the Ottoman Turks it was all about Jihad and the subjugation of non-Muslims in newly conquered territories. The persecution and discrimination against non-Muslims for the subjects of their realm was one of the founding principles of taheir rule and public policy.
That's not to mention that forcing Christianity on conquered subjects was one of the main goals of the British Empire.
My point is that it's irrelevant in terms of how history affects peoples' actions. You're engaging a detailed study of historical reality, but the masses don't do that. They have a narrative spun for them - generally no more than two or three sentences of complexity, max - and that's what matters.
The British Empire was vast and I can't come up with a definite answer for all the areas that were colonized by them but in my country "Egypt", they didn't do that. They however facilitated the work of western missionaries but it didn't achieve much with the Muslim locals and probably contributed to the outrage of Coptic clergy as these missionaries were snatching people from their congregation. Other than that, the British didn't really care about religion and their social order was constructed on race/class first basis and anything else second while for the Ottomans it was all about religion first and anything else second.